Find the Maximum/Minimum Value 2(x-3)^2+5

$2 {(x - 3)}^{2} + 5$

Step 1

Simplify each term.

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$f (x) = 2 ((x - 3) (x - 3)) + 5$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 2 (x (x - 3) - 3 (x - 3)) + 5$

Apply the distributive property.

$f (x) = 2 (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 5$

Apply the distributive property.

$f (x) = 2 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 2 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) + 5$

Move $- 3$ to the left of $x$ .

$f (x) = 2 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) + 5$

Multiply $- 3$ by $- 3$ .

$f (x) = 2 (x^{2} - 3 x - 3 x + 9) + 5$

Subtract $3 x$ from $- 3 x$ .

$f (x) = 2 (x^{2} - 6 x + 9) + 5$

Apply the distributive property.

$f (x) = 2 x^{2} + 2 (- 6 x) + 2 \cdot 9 + 5$

Simplify.

Multiply $- 6$ by $2$ .

$f (x) = 2 x^{2} - 12 x + 2 \cdot 9 + 5$

Multiply $2$ by $9$ .

$f (x) = 2 x^{2} - 12 x + 18 + 5$

Step 2

Add $18$ and $5$ .

$f (x) = 2 x^{2} - 12 x + 23$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 12}{2 (2)}$

Remove parentheses.

$x = - \frac{- 12}{2 (2)}$

Simplify $- \frac{- 12}{2 (2)}$ .

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$x = - \frac{2 \cdot - 6}{2 \cdot 2}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 2$ .

$x = - \frac{2 \cdot - 6}{2 (2)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 6}{2 \cdot 2}$

Rewrite the expression.

$x = - \frac{- 6}{2}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$x = - \frac{2 \cdot - 3}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = - \frac{2 \cdot - 3}{2 (1)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 3}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$x = - - 3$

Multiply $- 1$ by $- 3$ .

$x = 3$

Step 5

Evaluate $f (3)$ .

Replace the variable $x$ with $3$ in the expression.

$f (3) = 2 {(3)}^{2} - 12 \cdot 3 + 23$

Simplify the result.

Simplify each term.

Raise $3$ to the power of $2$ .

$f (3) = 2 \cdot 9 - 12 \cdot 3 + 23$

Multiply $2$ by $9$ .

$f (3) = 18 - 12 \cdot 3 + 23$

Multiply $- 12$ by $3$ .

$f (3) = 18 - 36 + 23$

Simplify by adding and subtracting.

Subtract $36$ from $18$ .

$f (3) = - 18 + 23$

Add $- 18$ and $23$ .

$f (3) = 5$

The final answer is $5$ .

$5$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(3, 5)$

Step 7

Find the Maximum/Minimum Value 2(x-3)^2+5

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